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A Universal Turing Machine

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The set of Turing machines forms a language: each string of the language is ... The set of all Turing Machines. is countable. Proof: Find an enumeration procedure ... – PowerPoint PPT presentation

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Title: A Universal Turing Machine


1
A Universal Turing Machine

2
A limitation of Turing Machines
Turing Machines are hardwired
they execute only one program
Real Computers are re-programmable
3
Solution
Universal Turing Machine
Attributes
  • Reprogrammable machine
  • Simulates any other Turing Machine

4
Universal Turing Machine
simulates any other Turing Machine
Input of Universal Turing Machine
Description of transitions of
Initial tape contents of
5
Tape 1
Three tapes
Description of
Universal Turing Machine
Tape 2
Tape Contents of
Tape 3
State of
6
Tape 1
Description of
We describe Turing machine as a string of
symbols We encode as a string of symbols
7
Alphabet Encoding
Symbols
Encoding
8
State Encoding
States
Encoding
Head Move Encoding
Move
Encoding
9
Transition Encoding
Transition
Encoding
separator
10
Machine Encoding
Transitions
Encoding
separator
11
Tape 1 contents of Universal Turing Machine
encoding of the simulated machine as
a binary string of 0s and 1s
12
A Turing Machine is described with a binary
string of 0s and 1s
Therefore
The set of Turing machines forms a language
each string of the language is the binary
encoding of a Turing Machine
13
Language of Turing Machines
(Turing Machine 1)
L 010100101, 00100100101111,
111010011110010101,
(Turing Machine 2)

14
Countable Sets

15
Infinite sets are either
  • Countable
  • or
  • Uncountable

16
Countable set
Any finite set
or
Any Countably infinite set
There is a one to one correspondence between
elements of the set and Natural numbers
17
Example
The set of even integers is countable
Even integers
Correspondence
Positive integers
corresponds to
18
Example
The set of rational numbers is countable
Rational numbers
19
Naïve Proof
Rational numbers
Correspondence
Positive integers
20
Better Approach
21
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26
Rational Numbers
Correspondence
Positive Integers
27
We proved the set of rational numbers is
countable by describing an enumeration
procedure
28
Definition
Let be a set of strings
An enumeration procedure for is a Turing
Machine that generates all strings of one
by one
and Each string is generated in finite time
29
strings
Enumeration Machine for
output
(on tape)
Finite time
30
Enumeration Machine
Configuration
Time 0
Time
31
Time
Time
32
Observation
If for a set there is an enumeration procedure,
then the set is countable
33
Example
The set of all strings is countable
Proof
We will describe an enumeration procedure
34
Naive procedure
Produce the strings in lexicographic order
Doesnt work strings starting with
will never be produced
35
Proper Order
Better procedure
1. Produce all strings of length 1 2. Produce
all strings of length 2 3. Produce all strings
of length 3 4. Produce all strings of length
4 ..........
36
length 1
Produce strings in Proper Order
length 2
length 3
37
Theorem
The set of all Turing Machines is countable
38
Enumeration Procedure
Repeat
1. Generate the next binary string of 0s
and 1s in proper order 2. Check if the string
describes a Turing Machine if
YES print string on output tape if
NO ignore string
39
Uncountable Sets

40
A set is uncountable if it is not countable
Definition
41
Theorem
Let be an infinite countable set The
powerset of is uncountable
42
Proof
Since is countable, we can write
Elements of
43
Elements of the powerset have the form

44
We encode each element of the power set with a
binary string of 0s and 1s
Encoding
Powerset element
45
Lets assume (for contradiction) that the
powerset is countable.
Then we can enumerate the
elements of the powerset
46
Powerset element
Encoding
47
Take the powerset element whose bits are the
complements in the diagonal
48
New element
(birary complement of diagonal)
49
The new element must be some of the powerset
50
Since we have a contradiction
The powerset of is uncountable
51
An Application Languages
Example Alphabet
The set of all Strings
infinite and countable
52
Example Alphabet
The set of all Strings
infinite and countable
A language is a subset of
53
Example Alphabet
The set of all Strings
infinite and countable
The powerset of contains all languages
uncountable
54
Languages uncountable
Turing machines countable
There are more languages than Turing Machines
55
Conclusion
There are some languages not accepted by Turing
Machines
(These languages cannot be described by
algorithms)
56
Languages not accepted by Turing Machines
Languages Accepted by Turing Machines
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